HYDROCARBON LIQUID FLOW CALIBRATION SERVICE 

NIST Special Publication 250-1039 

T. T. Yeh, Jesús Aguilera, and John D. Wright 

Fluid Metrology Group 

Process Measurements Division 

Chemical Science and Technology Laboratory 

National Institute of Standards and Technology 

Gaithersburg, MD 20899 
Hydrocarbon Liquid Flow Calibration Service

Table of Contents 

 Abstract .............................................................................................1 

1.0 Introduction............................................................................................................. 1

2.0 Description of Measurement Services .................................................................... 2 

3.0 Procedures for Submitting a Flowmeter for Calibration......................................... 3 

4.0 Description of the Hydrocarbon Liquid Flow Standard .......................................... 4 

4.2 The Equation for Flow at the Meter Under Test..................................................... 7 

5.0 Uncertainty Analysis Overview............................................................................ 11 

5.1 Techniques for Uncertainty Analysis.................................................................... 12 

5.2 Measured Quantities and Their Uncertainties....................................................... 13 

5.3 Propagating Components of Uncertainty.............................................................. 19 

5.4 Combined and Expanded Uncertainty .................................................................. 20 

6.0 Uncertainty of Meter Under Test.......................................................................... 21 

7.0 References............................................................................................................. 22

Appendix A: Volume Balance Method ......................................................................... 23 

Appendix B: Supporting Data for Uncertainty Analysis............................................... 28 

Appendix C: Sample Calibration Report....................................................................... 38 

ii
Abstract 

This document provides a description of a 2 liter hydrocarbon liquid flow calibration standard 

operated by the National Institute of Standards and Technology (NIST) Fluid Metrology Group 

to provide flowmeter calibrations for customers. This facility uses a passive piston prover 

technique where fluid is driven by pumps while the measuring piston is pushed by the fluid over 

a measured length during a measured time. This facility is presently operated using MIL-C7024C fluid,∗

 but other fluids could be used to offer a different range of fluid properties. The 

facility has uncertainty of ± 0.01 % (k=2 or approximately 95 % confidence level) over the flow 

range 0.19 L/min to 5.7 L/min (0.05 gal/min to 1.5 gal/min). 

In this document, we provide an overview of the hydrocarbon liquid flow calibration service and 

the procedures for customers to submit their flowmeters to NIST for calibration. We derive the 

equation for calculating flow at the meter under test, including the corrections for storage effects 

caused by changes in fluid density and connecting volume (due to temperature changes). Finally 

we analyze the uncertainty of the flow standard, give supporting data, and provide a sample 

calibration report. 

Key words: calibration, flow, connecting volume, flowmeter, hydrocarbon liquid, flow standard, 

uncertainty 

1.0 Introduction 

Flow measurement units are derived from the SI base units. Therefore, the paths taken to realize 

of flow measurement standards vary and depend upon such issues as the properties of the fluid(s) 

to be measured. Realization methods at NIST are always derived from fundamental 

measurements such as mass, length, time, and temperature, typically by accounting for the 

transfer of a known mass or volume of fluid over a measured time interval under approximately 

steady state conditions of flow, pressure, and temperature at the meter under test (MUT). Such 

flow metrology facilities are known as “primary flow standards”, and by definition [1], they are 

facilities capable of determining flow, at specific, quantifiable uncertainty levels, without being 

calibrated for the unit of flow being realized. 

For many years, NIST has provided flowmeter calibrations in hydrocarbon liquid flows up to 

1500 L/min using a dynamic gravimetric flow standard that has uncertainty of 0.12 %. Recently, 

we have worked to replace the gravimetric standard with new, lower uncertainty facilities that 

also have a higher level of automation. This document describes the first of these new systems, a 

2 liter Hydrocarbon Liquid Flow Standard that covers flows up to 5.7 L/min with uncertainty of 

0.01%. 

The 2 liter Hydrocarbon Liquid Flow Standard (2 L HLFS) consists of a fluid source (e.g., tank), 

a long test section that provides stable thermodynamic conditions and a fully developed flow 

profile, and a system for timing the displacement of a quantity of the fluid. The flow measured 

by the primary standard is computed along with the average of the flow indicated by the MUT

during the collection interval. All of the quantities measured in connection with the calibration 

standard (i.e., temperature, pressure, density, time, etc.) are traceable to established national 

standards. 

 

∗ Also know as Stoddard solvent – a surrogate liquid for JP-4 and JP-5 jet fuels.

1
NIST calibrates hydrocarbon liquid flowmeters to provide traceability for flowmeter 

manufacturers, secondary flow calibration laboratories, and flowmeter users. We calibrate a 

customer’s flowmeter and deliver a calibration report that documents the calibration procedure, 

the calibration results, and their uncertainty on a fee-for-service basis. The customer may use the 

flowmeter and its calibration results in different ways. The flowmeter is often used as a transfer 

standard to compare the customer’s primary standards to the NIST primary standards so that the 

customer can establish traceability, validate their uncertainty analysis, and demonstrate 

proficiency. Customers with no primary standards use their NIST calibrated flowmeters as 

working standards or reference standards in their laboratory to calibrate other flowmeters. The 

Report of Calibration is the property of the customer and NIST considers the results of 

calibrations to be proprietary information of the customer. 

Operators of primary flow standards seek to validate the claimed uncertainties of their standards 

by establishing and maintaining the traceability of calibration results to the SI. One complete 

way to establish traceability involves the use of proficiency testing techniques, which quantify 

the traceability of a facility’s results using a set of flow standards maintained by a National 

Metrology Institute (NMI) [2]. Alternatively, establishing traceability can be achieved through 

assessment of individual facility components and analyzing their respective contributions to the 

calibration process. An extensive component analysis of NIST’s HLFS has been performed and 

is described in following sections of this document using the component analysis method. 

2.0 Description of Measurement Services 

Customers should consult the web address www.nist.gov/fluid_flow to find the most current 

information regarding our calibration services, calibration fees, technical contacts, and 

flowmeter submittal procedures. 

NIST uses the hydrocarbon liquids standards described herein to provide hydrocarbon liquids 

flowmeter calibrations for flows between 0.19 L/min and 5.70 L/min [3]. The facility can be 

used at flows as low as 0.05 L/min or as high as 7 L/min, but calibrations below 0.19 L/min or 

over 5.7 L/min should be discussed with the technical contacts before a flowmeter is submitted 

for such calibrations. 

The liquid used for calibrations is normally MIL-C-7024C fluid (also known as Stoddard or Type 

2 solvent), but calibrations using other fluids are possible. The hydrocarbon liquid temperature 

during the calibration is nominally 22.2 ± 0.5 o

C. Readily available fittings for the installation of

flowmeters for calibration are Swagelok∗

, A/N 37 degree flare, and national pipe thread (NPT). 

Meters are tested if the flow range and piping connections are suitable, and have precision 

appropriate for calibration with the NIST flow measurement uncertainty. The vast majority of

flowmeters calibrated in the hydrocarbon liquids flow calibration service are turbine flowmeters 

because these are presently regarded as the best candidates for transfer and working standards by 

the hydrocarbon liquid flow metrology community. Meter types with calibration instability 

significantly larger than the primary standard uncertainty are generally not calibrated with the 

 

∗ Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such 

identification does not imply recommendation or endorsement by the National Institute of Standards and 

Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the 

purpose.

2
NIST standards since the job can be done with acceptable uncertainty more economically by 

other, commercial labs. 

A normal flow calibration performed by the NIST Fluid Metrology Group consists of six flows 

spread over the range of the flowmeter. A flowmeter is normally calibrated at 10 %, 20 %, 35 %, 

50 %, 75 %, and 100 % of its full scale. At each of these flow set points, five (or more) flow 

measurements are made consecutively. The same set point flows are tested on a second occasion, 

but the flows are tested in decreasing order instead of the increasing order of the first set. 

Therefore, the final data set consists of ten (or more) flow measurements made at six flow set 

points, i.e., 60 individual flow measurements. The sets of five measurements can be used to 

assess repeatability, while the sets of ten can be used to assess reproducibility. For further 

explanation, see the sample calibration report in Appendix C of this document. Variations on the 

number of flow set points, spacing of the set points, and the number of repeated measurements 

can be discussed with the NIST technical contacts. However, for data quality assurance reasons, 

we rarely will conduct calibrations involving fewer than three flow set points and two sets of 

three flow measurements at each set point. 

The Fluid Metrology Group prefers to present flowmeter calibration results in a dimensionless 

format that takes into account the physical model for the flowmeter type. The dimensionless 

approach facilitates accurate flow measurements by the flowmeter user even when the conditions 

of usage (liquid type, temperature) differ from the conditions during calibration. Hence for a 

turbine calibration, the calibration report will present Roshko number vs Strouhal number [4]. In 

order to calculate the uncertainty of these flowmeter calibration factors, we must know the 

uncertainty of the standard flow measurement as well as the uncertainty of the instrumentation 

associated with the MUT (normally frequency and temperature instrumentation). NIST-owned 

and controlled instruments (temperature, etc.) are used as part of the test of a customer’s meter 

since these have established uncertainty values based on calibration records maintained as part of 

the our Quality System. Such information is not available for the customer’s instrumentation. 

Use of the customer’s instrumentation for a calibration or special test requires specific 

arrangements to be made for such operations. Calibration of customer’s ancillary 

instrumentation is not part of the calibration procedures described here and would require that 

separate arrangements be made. 

3.0 Procedures for Submitting a Flowmeter for Calibration 

The Fluid Metrology Group follows the policies and procedures described in Chapters 1, 2, and 

3 of the NIST Calibration Services Users Guide. These chapters can be found on the internet at 

the following addresses: 

http://ts.nist.gov/ts/htdocs/230/233/calibrations/Policies/policy.htm, 

http://ts.nist.gov/ts/htdocs/230/233/calibrations/Policies/domestic.htm, and 

http://ts.nist.gov/ts/htdocs/230/233/calibrations/Policies/foreign.htm. 

Chapter 2 gives instructions for ordering a calibration for domestic customers and has the subheadings: A.) Customer Inquiries, B.) Pre-arrangements and Scheduling, C.) Purchase Orders, 

D.) Shipping, Insurance, and Risk of Loss, E.) Turnaround Time, and F.) Customer Checklist. 

Chapter 3 gives special instructions for foreign customers. The web address 

www.nist.gov/fluid_flow has information more specific to the flow calibration service, including 

the technical contacts in the Fluid Metrology Group, fee estimates, and turn around times. 

3
4.0 Description of the Hydrocarbon Liquid Flow Standard 

Piston prover systems have long been accepted as primary flow calibrators for both gas and 

liquid flowmeters [5, 6]. In its most basic form, the piston prover consists of a circular cylinder 

of known internal diameter, which encompasses a sealed piston. This piston strokes through 

measured lengths, at a constant speed, to produce a volumetric flow. The cross sectional area of 

the cylinder multiplied by the length traveled gives a volume. The volumetric flow is calculated 

by dividing the volume by the time it took the piston to traverse the length. Temperature 

measurements at key locations are used to assess the changes in fluid density and connecting 

volume that occur between the start and stop times, allowing corrections for storage effects. The 

coefficient of thermal expansion of the calibration fluid is known, hence the volumetric flow in 

the cylinder can be converted to the volumetric flow at the test section where the meter under 

test (MUT) is located (or the mass flow can be determined from the fluid density). The output of 

the meter under test is acquired along with the necessary piston measurements so that average 

flows from the MUT and the flow standard can be compared. 

Two types of piston arrangements are generally used in these systems. An active piston can both 

drive and measure a volumetric flow out of the cylinder (like a syringe), while a passive piston 

operates via pumped fluid, which drives the piston through the calibration stroke to determine 

the volumetric flow. The NIST HLFS employs a passive piston. 

Table 1. Nominal Characteristics of NIST’s 2 liter Hydrocarbon Liquid Flow Standard. 

 ∆Vp , [cm3

] 560 to 1530**

D , [cm] 7.62 

d , [cm] 2.54 

L , [cm] 55 

Lc , [cm] 14 to 40**

c t , [s] 15 to 180**

uC , [%] 0.005 

uk =2 , [%] 0.01 

Q , [L/min] 0.19 to 5.7 

 

** Specific values are selected to conform with those used in the following uncertainty analyses.

4
The 2 L HLFS was constructed by Flow Dynamics Inc, in Scottsdale, AZ.*

 Upon receipt at NIST, 

we performed the calibrations of length, time, temperature, and fluid properties necessary to 

make the system directly traceable to NIST standards and made an uncertainty analysis and those 

efforts are documented herein. 

Figure 1. Photograph of the small Liquid Flow Calibrator.

The HLFS is operated in a closed loop mode. As shown schematically in Figure 2, the 

hydrocarbon liquid is pumped from the reservoir tank by two pumps. Depending on the required 

flow, some of the pumped liquid is returned to the reservoir directly. The remaining, required 

flow is passing through the piston-valve assembly and then to the test section, where the MUT is 

located. After the MUT, the entire flow is returned to the reservoir to complete the flow loop. 

The HLFS uses two three-way valves to allow the pumped liquid to move the piston back and 

forth in a bi-directional mode while maintaining unidirectional flow through the MUT.

Calibration data is acquired with the piston moving in either direction. 

 

* Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such 

identification does not imply recommendation or endorsement by the National Institute of Standards and 

Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the 

purpose. 

5
Figure 2. A sketch of the calibrator with the piston stroking left. The locations of 

temperature sensors are shown. 

The operation of the four-way diverter valve is demonstrated schematically in Figures 2 and 3. 

The calibration interval begins as soon as flow conditions reach steady state and the piston 

accelerates to constant velocity (shortly after the piston begins travel in one direction). During 

the transition period when the piston changes direction, both of the three-way valves are set so 

that all three ports are open, thus preventing any hydraulic ram effects. During the change in 

direction, flow stops at the MUT, and it is necessary to wait for steady state conditions before 

beginning to collect calibration data. The meter output averaging is stopped before the piston 

reaches the transition period. 

To allow a range of flows, a computer-controlled stepping-motor drives both pumps. The 

capacity of the small pump is about one tenth the larger one. At lower flow conditions, the flow 

from the large pump is bypassed to the reservoir tank. 

Steady state temperature conditions are achieved by cycling the piston back and forth at the 

intended calibration flow, which promotes the mixing of the fluid. A chilled water heat 

exchanger controlled by a feedback temperature sensor is used to remove heat from the fluid 

added by friction and the pumps. Thermal equilibrium is further enhanced by forcing the 

calibration fluid through an outer cylindrical jacket that encloses the cylinder and piston. The 

liquid flow from the pumps is directed into this outer jacket before entering the four-way diverter 

valve. Two benefits of using this outer jacket are: (1) better temperature uniformity among 

different parts of the fluid and the prover, and (2) pressure balance inside and outside of the 

prover, preventing a pressure gradient from affecting the diameter of the cylinder. The calibrator 

uses twelve temperature sensors to determine fluid properties throughout the flow loop. The 

locations are indicated in Figure 2. Furthermore, the calibrator is located in a temperaturecontrolled room where the air temperature is maintained at 22.2 ±1o

C. These considerations are 

6
important for minimizing changes in density in the connecting volume between the cylinder and 

the MUT and the resulting corrections for storage effects. 

(b) Transition from Left to Right Stroke

(d) Transition from Right to Left stroke

(a) Piston Stroking Left

(c) Piston Stroking Right

(b) Transition from Left to Right Stroke

(d) Transition from Right to Left stroke

(a) Piston Strok ston Stroking Left

(c) Piston Stroking Right

Figure 3. Diagrams of piston and three-way valves reversing piston directions. 

The position of the piston along the cylinder length is measured with two, redundant linear 

encoders, one on each of the supporting shafts attached to either side of the piston. The encoders 

output nominally 50 square wave pulses per mm of length traveled. The time for the piston to 

travel a given length is measured by counting pulses from two redundant 1 MHz timers (see 

section 5.2). 

Leaks of calibration fluid can be visually detected. There are two o-ring seals on the piston, one 

on the leading edge, another on the trailing edge. Fluid that leaks past either of these seals flows 

through machined holes in the piston and piston shafts to the shaft ends where it will drip out. 

Leaks past the piston shaft seals and from pipe fittings are easy to detect by eye. 

4.2 The Equation for Flow at the Meter Under Test 

The HLFS measures the volumetric flow of fluid through the MUT from the fluid discharged by 

the piston-cylinder assembly over a precisely measured time period. Since leaks or temperature 

and pressure changes may cause the total mass in the pipe that connects the piston-cylinder 

assembly to the MUT (the connecting volume) to change during the calibration, the flow 

through this meter may differ from the flow exiting the cylinder (storage effects). If no leaks are 

7
found, if corrections for temperature and pressure effects have been made, and if the fluid 

volume change in the connecting pipe is accounted for, the volumetric flow at the MUT can be 

obtained. Most previous prover analyses do not include corrections for fluid mass change in the 

connecting volume and fluid density is assumed constant during the calibration (they are treated 

as uncertainties instead). However, the connecting volume effect could contribute a significant 

error in the volumetric flow determination. 

Different methods can be used in accounting for the connecting volume effect. The most 

common correction technique, which is based on the mass conservation method, is given below.

A different method based on volume balance is given in Appendix A. 

Conservation of Mass

The operational principle of the piston prover is the conservation of mass. The general equation 

for the conservation of mass is: 

+ ⋅ = 0

∂

∂

∫ ∫ dV U dA

t V A

r r ρ ρ (1) 

where ρ is the fluid density; ∂ is the partial derivative with respect to time; V is a control 

volume used to assess the specified mass balance; and U

/ ∂t v

 is the vectorial velocity across the 

vectorial area element, dA

v

, of the control surface (surrounding the control volume) with the 

positive direction outward, such that the positive U dA v v ⋅ product denotes outflow. Equation (1) 

states that mass is neither created nor destroyed. For this system, that is, the initial fluid mass in 

the control volume, at the beginning of the calibration time interval, equals the final fluid mass at 

the end of the calibration, 

M Fi = M Ff (2) 

where the subscripts i and denote the initial and final conditions. For a piston prover 

calibrator as shown in Figure 4, the initial fluid mass is 

f

M Fi

M Fi = M Pi + M CVi + M MUTi (3) 

and the final fluid mass is M Ff

M Ff = M Pf + M CVf + M MUTf + M L (4) 

In (3) and (4), the subscripts P , , and denote the piston, connecting volume, and MUT,

respectively. is any mass leaking from the system between the prover and MUT. The 

leakage is usually apparent and leaks are repaired before the calibrator can be used. Therefore, 

the leakage term is hereafter assumed to be zero. 

CV MUT

M L

M L

8
Figure 4. Mass conservation as applied to an incompressible piston system. 

From (2) ~ (4), we have the mass of fluid passing through the MUT as: 

∆M MUT = M MUTf − M MUTi = ∆M P − ∆M CV (5) 

where 

∫ ∫ ∆ = − = −

VPi VPf

P Pi Pf M M M ρ dV ρ dV (6) 

is the total mass displaced by the piston during the time period, , and ct

∆M CV = M CVf − M CVi (7) 

is the total mass difference in the connecting volume during the same period, t . When the fluid 

density in the piston is constant during the calibration time and equal to 

c

ρ P , (6) simplifies to: 

Pi Pi Pf Pf P P

V V

M P dV dV V V V

Pi Pf

∆ = − = − = ∆ ∫ ∫ ρ ρ ρ ρ ρ (8) 

where is the total volume displaced by the piston moving along the cylinder in the 

calibration in time, t . 

∆VP

c

Connecting Volume Effects

In general, the fluid density is a function of pressure and temperature. However, in this liquid 

calibrator, the effect of the pressure variation is negligible, since the pressure is nearly constant 

throughout the system, the maximum operating pressure is 200 kPa, and the modulus of 

elasticity of the calibration liquid is large ( ~ 2 GPa). Therefore, in this analysis, the liquid 

density is assumed to be a function of the temperature only. Similarly, the structural length and 

volume of the connecting piping is assumed to be a function of temperature only (i.e., no 

9
structural deformation in the pipe due to liquid pressure). For a small temperature change, ∆T , 

the following linear relationships are assumed: 

( )

( )

V V ( ) T

L L T

T

s

s

= + ∆

= + ∆

= − ∆

α

α

ρ ρ α

1 3

1

1

0

0

0

(9) 

where α is the thermal expansion coefficient for the liquid density, and α s is the linear 

expansion coefficient for the connecting pipe. Thus, any mass change in the connecting volume 

can be expressed as: 

( )

( ) (

( )( ) CV CV S CP CV

CV CV s CP CV CV CV

CV CV CV CV

CV CV

V T T

V T V T

V V

M V

= ∆ − ∆

= ∆ − ∆

= ∆ + ∆

∆ = ∆

ρ α α

ρ α ρ α

ρ ρ

ρ

3

3 )

(10) 

where ρCV is the average fluid density in the connecting volume; V is the average connecting 

volume; is the temperature rise of the connecting pipe between the initial and final times 

for the calibration interval; and is the temperature rise of the fluid in the connecting 

volume between the initial and final times. 

CV

∆TCP

∆TCV

Substituting (8) and (10) into (5), the fluid mass through the MUT in the calibration interval, t , 

becomes 

c

( )( ) ∆M MUT = ρ P ∆VP − ρ CV VCV 3α s∆TCP −α∆TCV (11) 

Dividing (11) by the fluid density in the MUT, ρ MUT , and the calibration interval, , the 

average volumetric flow rate through the MUT, Q , becomes: 

ct

MUT

( )( )

MUT c

P P CV CV s CP CV

MUT c

MUT

MUT t

V V T T

t

M

Q

ρ

ρ ρ α α

ρ

∆ − ∆ − ∆ = ∆ = 3

(12) 

From (9), the density ratios can be expressed as: 

P,MUT

MUT

P = −α∆T

ρ

ρ 1 and CV ,MUT

MUT

CV = −α∆T

ρ

ρ 1 (13) 

where is the temperature difference between the fluid in the piston prover and the fluid 

in the MUT and is the average temperature difference between the fluid in the 

∆TP,MUT

∆TCV ,MUT

10
connecting volume and that in the MUT. Substituting equation (13) into (12), the volumetric 

flow rate equation becomes: 

( )

( ) 



 





+ ∆ ∆ − ∆

∆ − ∆ − ∆ − ∆ =

CV ,MUT CV S CP CV

P P,MUT CV S CP CV

c

MUT T V T T

V ( T ) V T T

t

Q

α α α

α α α

3

1 1 3

(14) 

The last term in (14) is a second order expansion quantity which can be neglected based on a 

magnitude analysis. Hence, the final equation for volumetric flow rate becomes: 

[ ] ( ) P P,MUT CV CV S CP

c

MUT V ( T ) V T T

t

Q = ∆ 1−α∆ + α∆ − 3α ∆ 1

(15) 

Equation (15) is the basis equation for calculating flow from the HLFS. The first term in the 

parentheses represents the flow from the piston volume with a thermal expansion correction. The 

second term in the parentheses accounts for storage effects in the connecting volume, 1) due to 

density changes in the fluid caused by temperature changes and 2) due to changes in the volume 

of the tubing holding the fluid. 

From (15), it can be seen that the fluid density does not directly affect the calculation of the 

volumetric flow rate; however, any change in fluid density in the connecting volume is given by 

the terms −α ∆TP,MUT and VCVα ∆TCV . Thus the value of liquid density is only important if a 

mass flow, ρ MUTQMUT , is needed. 

An alternate method, using volume balance, for correcting fluid volume in the connecting 

volume gives a same result as that of the equation (15). The volume balance method is given in 

Appendix A. 

5.0 Uncertainty Analysis Overview

The uncertainty components of the Hydrocarbon Liquid Flow Standard are discussed in detail in 

following sections. As seen in (15), they include the uncertainty of the following elements: 

a. the displaced prover volume, ∆VP , and the linear thermal expansion coefficient for the 

piping, α S , 

b. several temperature differences (∆TP,MUT , ∆TCP , ∆TCV ), 

c. the connecting volume, V , CV

d. the calibration interval, tc, and 

e. the thermal expansion coefficient of the liquid density, α . 

Some component uncertainties listed above and in Equation (15) could not be measured directly. 

Their uncertainties are estimated from the uncertainties of their source measurements using the 

first order uncertainty propagation method to be discussed below. Figure 5 shows a graphic 

representation tree of the uncertainty analysis for the HLFS. 

 

11
`Figure 5. Graphic representation of the uncertainty analysis. 

5.1 Techniques for Uncertainty Analysis 

Here we follow the guidelines for evaluating and expressing uncertainty provided in 

NIST TN 1297 [7], the ISO Guide [8], and elsewhere [9]. In general, if a measurement quantity, 

y , is a function of variables , i x

y f ( x , x ,....., x ) = 1 2 n (16) 

its first-order Taylor series approximation is, 

∑ ∂

∂ =

i

i

i

dx

x

y dy (17) 

Thus, the propagation of uncertainty yields 

12
r u( x )u( x ) x

y

x

y u ( x ) xy

u( x ) x

y u ( y )

ij i j

n

j i j

n

i i

i

n

i i

n

i

i

i

c

∑ ∑ ∑

∑

= +

−

= =

=

















∂

∂

















∂

∂

+ 















∂

∂ =

































∂

∂ =

1

1

1

2

1

2

2

1

2

2

(18) 

where is the combined standard uncertainty of the measurement result , is the 

standard uncertainty of the variable , the partial derivatives 

u ( y ) c y ( i u x )

i x i ∂y / ∂x are the dimensional 

sensitivity coefficients of on , and is the cross correlation coefficient between variables 

 and . An alternative form of (18), which expresses the uncertainty propagation in a 

dimensionless form, is shown below and it is often more useful. 

i x y ij r

i x j x

i i i j i j ij x x

n

j i

x ,y

n

i

x x ,y

n

i

y x ,y u ∑c u ∑c ∑c r u u

= +

−

= =

= +

1

1

1

2

1

2 2 2 (19) 

In (19), is the combined dimensionless standard uncertainty of the measurement 

result , are the dimensionless sensitivity coefficients of on , and 

 is the dimensionless uncertainty of the variable . Equation (19) is used here to 

estimate the combined uncertainty of the measurement. In many cases, the uncertainty of 

could not be measured directly. For those cases, the same uncertainty propagation given by (19) 

is used for a sub-measurement process to estimate the combined uncertainty of the 

sub-measurement. This process is propagated throughout all the measurement components 

needed until the desired measured quantities are obtained. 

u u ( y ) / y y = c

c ( y / x x ,y i = ∂ ∂

( )i i x / x

y

u

)x / y i i i x y

x u i = i x

i x

According to [8], the sources of uncertainty used in assessing the combined standard uncertainty 

of the measurement process can be classified according to two types: Type A - those which are 

evaluated by statistical methods, and Type B - those which are evaluated by other means. 

Following this convention, each measured quantity has been classified accordingly as a or 

. 

A u

B u

5.2 Measured Quantities and Their Uncertainties 

As noted above, the uncertainty propagation can be calculated based on (15), but it is worth 

remembering that the flow and the related components in (15) are not directly measured. To

obtain the uncertainty of this result, its relationship to other measured quantities is required. In 

this section, the relationships are given. 

The Displaced Volume

The piston prover volume, , can be calibrated by: (a) water draw method, (b) using a master 

flowmeter, or (c) by using the dimensions of its diameter and the distance traveled by the piston. 

Here, the diameter/traveled-distance method is used to determine the prover volume. The piston 

displacement is measured using optical encoders. In terms of the measured quantities, ∆V

∆VP

P is,

13
( ) ( )

E

E

P E K

D d n L D d V

4 4

2 2 2 2 − ∆ = − ∆ = π π (20)

where D and d are the piston prover diameter and piston shaft diameter, respectively; ∆LE is 

the encoder (or piston) displacement length, is the encoder constant (in pulses/cm), and 

is the total number of pulses produced by the encoder during the calibration stroke. In our 

system, the value may be selected by software or by the operator, to ensure that the prover is 

operating at the uncertainty asserted in this analysis. The prover unit was disassembled so the 

shaft and the cylinder diameters could be measured accurately. The shaft diameter was measured 

using an outside diameter micrometer (Mitotoyo Model MDC-2”JT). Two standard blocks, one 

slightly smaller and one slightly larger than the shaft diameter, calibrated by the NIST Precision 

Engineering Division were used to calibrate the micrometer and establish traceability. Table B3 

in Appendix B shows the measurement results and their uncertainties. The cylinder inside 

diameter was measured using a three-legged inside diameter micrometer (Mitutoyo, Model IT005D). Two standard rings, one slightly smaller and one slightly larger than the cylinder inside 

diameter, calibrated by the NIST Precision Engineering Division, were used to calibrate the 

inside micrometer. Table B4 in Appendix B shows the data for the inside diameter and their 

uncertainties. 

KE E n

E n

In bi-directional piston prover calibrators, the piston diameter is the same, regardless of the 

direction of piston travel. However, the shafts that are attached to both sides of the piston may 

have slightly different diameters. Similarly, if multiple linear encoders, as is the case here, 

monitor the displacement of the piston shafts, they may have slightly different encoder constants. 

Therefore, for bi-directional provers, the measurement characteristics in one direction of piston 

travel may be different from the other. Nonetheless, because the shaft diameter assessment 

process is based on the entire shaft on both sides and the times measured from both encoders are 

used for either direction (see time measurement below), it is expected that the uncertainty for 

each direction will be the same. 

In this system, the encoders are Mitutoyo AT2N-600. A stabilizer is built into the detector to 

ensure a stable signal output. The encoder detector head is trued to the glass scale inside its 

housing and fixed by positioning plates. The thermal expansion of the encoders is given by, 

( ) LE = LER +α E ∆TER 1 and ( ) KE = KER −α E ∆TER 1 (21)

where and are the encoder length and encoder constant at the reference temperature, 

respectively; is the temperature rise from the reference temperature, and 

LER KER

∆TER α E is the linear 

expansion coefficient of the encoder. The encoder constant was calibrated using a reference 

laser interferometer (HP Model 10565A) borrowed from the Precision Engineering Division. The 

calibration was made by comparing the laser interferometer output with the inline output of the 

encoder pulses. Table B5 in Appendix B shows the measurement results and their uncertainties. 

The diameters of the prover assembly also change with the temperature following these relations, 

( ) D = DR 1+α S ∆TDR and ( ) d = d R 1 +α S ∆TdR (22) 

14
where and are the reference temperature diameters of the prover and of the piston shaft, 

respectively (see Tables B3 and B4), and 

DR R d

∆TDR and ∆TdR are the temperature rise with respect to 

their reference temperature, respectively. 

Temperature Measurement

The uncertainty of the temperature measurements made throughout the prover will contribute to 

the uncertainty of the calibrator. Thermistors are used for all temperature measurements, with 

twelve of them placed at various locations along the liquid flow path. At locations deemed 

critical, the system has duplicate sensors to improve measurement accuracy. The locations of the 

temperature sensors in the system are shown in Figure 2. TAthena

1B

is located near the heat 

exchanger and is used by the temperature controller; T and T are located at prover exit-1, 

and and T are at prover exit-2; T and T are located at the exit of the four-way 

value, and T are placed along the length of the connecting volume; T and 

are immediately upstream of the MUT. 

1A

T2A 1B

T

2B CV1A CV

CV 2 CV 3 MUTA TMUTB

The model used for the reduction of the various temperatures in the system affects the 

uncertainty of the calibrator results. At initial and final conditions, the average connecting 

volume fluid temperature, T , is assumed to be the average value of the five temperature 

readings made along the fluid path. 

CV

TCV = ( ) TP +TCV1 +TCV 2 +TCV 3 +TMUT / 5 (23)

In the above equation, T is the average prover temperature (depending on the 

piston travel direction, 1 or 2); 

P = (TPiA +TPiB / 2

i =

)

TCV1 = (TCV1A +TCV1B )/ 2 is the average temperature at exit 

of the four way valve; and TMUT = (TMUTA + TMUTB ) / 2 is the temperature at the MUT. This 

average temperature model results in an uncertainty for the average temperature. This additional 

uncertainty in the average temperature is given by. 

Emax = the maximum spatial variation of the fluid temperature. (24) 

Typical temperature variation in the connecting volume is shown in Figure 6. This and other test 

data show that the maximum temperature variation among the sensor locations is 0.1 K or less. 

The thermistors are periodically calibrated in an isothermal bath by comparing their response to 

that of a standard thermometer calibrated by the NIST Thermometry Group. The four calibration 

coefficients and the temperature uncertainty for each thermistor are obtained using a linear 

regression method. The uncertainty in the reference temperature, which is 0.002 K, is also 

classified as a Type B uncertainty for each sensor. An additional uncertainty is obtained by 

comparing the temperature output of the calibrated sensors with the reference temperature in the 

isothermal bath. The results from a sample calibration are shown in Tabel B1 in Appendix B. The 

average difference is assigned as the Type B uncertainty and the standard deviation of the 

difference is assigned as a Type A uncertainty. The root-sum-square of the uncertainties from the 

15
reference sensor, data regression, and temperature test, is the combined uncertainty of each 

sensor. In the worst case, the sensors have = 0.041 K and = 0.031 K. A u B u

295.5

295.6

295.7

295.8

TP1ATP2A

TP1BTP2B

TCV1A

TCV1B

TCV2 TCV3 

TUU4A 

TUU4B

TAir

TAthena

Temperature sensors

T [K]

Figure 6. Temperature measurements made by the 12 sensors in the HLFS on two different 

occasions. Temperature uniformity along the flow path is less than 0.1 K. 

The Connecting Volume

The connecting volume is modeled using the following equation: 

4 2 V d l / CV = π CV CV (25) 

In (25), is the averaged internal diameter of the connecting pipe and l is its length. There 

is significant uncertainty associated with the estimation of the quantities needed to precisely 

compute the connecting volume: piping inside diameters, piping lengths, internal volumes of the 

valves and elbows, the dead volume in the prover, the extra connecting volumes associated with 

the piping used for different MUTs, etc. However, as shown in Table 2 below, the sensitivity of

the connecting volume determination on the computed result for the volumetric flow through the 

MUT is quite small. The large uncertainty of 11.6 % in the connecting volume will result only 

5.8×10

dCV CV

-5 % of flow uncertainty. This is because the change in density of the fluid in the 

connecting volume during a flow measurement is small (the temperature profile is quite stable) 

and because the connecting volume is small in size compared to the volume swept out by the 

piston. Thus, the flow measurement uncertainty is relatively insensitive to the rather large 

uncertainty in the connecting volume size. Furthermore, the uncertainties of the dimension 

changes due to thermal expansion are even smaller and are neglected. 

16
Time of Piston Displacement

The uncertainty of the measurements of time can be separated into two parts: a) that due to the 

reference clock (including calibration errors and temperature effects) and b) that due to 

quantization errors. The time base oscillators are periodically calibrated by two reference 

counters that are traceable to NIST time and frequency standards. Table B7, Appendix B shows 

sample calibration results for the two oscillators in the HLFS. The Type A uncertainty of the 

HLFS oscillators used in this uncertainty analysis is 0.31 μHz/Hz and the Type B uncertainty is 

1.55 μHz/Hz. These oscillator calibration uncertainties are larger than the quantization 

uncertainties discussed below.

Figure 7 illustrates that the quantization error for a generic timed interval will be smaller than or 

equal to one time reference unit (± ∆t). The true time, ttrue, is marked by the start and stop times, 

t1 and t3. The data acquisition system obtains the measured time, tmeas, by counting the number of 

rising edges, n, from the reference clock (nominally 1 MHz in this case) between t2 and t4 and 

multiplying n by the reference time unit (1 μs). The timing errors at the start and end of the 

measurement (δs and δe) can each be between zero and one time unit in magnitude. The resulting 

difference between the true and measured times is ± one time reference unit. The start and end 

timing errors each have rectangular probability distributions and the difference between them

(the quantization error) has a standard uncertainty of ∆t 6 . Although the example uses rising 

edges for triggering, this analysis is equally valid for falling edges. It does assume that no pulse 

is missed by the counter. 

Figure 7. A sketch showing how reference clock pulses are used to time a generic interval and 

how the timing procedure leads to an quantization error of ± one time reference unit (± ∆t). 

Figure 8 illustrates the application of the reference clock to measurement of the time for the 

piston to travel a selected distance, as indicated by a selected number of pulses output by one of 

the encoders. The figure also illustrates the measurement of the frequency output by a MUT.

Once the test conditions have reached steady state (at time t0), the next rising edge output by the 

encoder is used to commence the counting of rising edges from the reference clock. After the 

predetermined number (nE) of encoder pulses has been registered by a counter, the counting of 

reference clock pulses is stopped, and the total (n) is multiplied by ∆t to obtain the time required 

17
by the piston to travel the prescribed distance (tEmeas). As for the generic case, this time has a 

standard uncertainty due to quantization of ∆t 6 . 

The frequency of the flowmeter output, fT, is calculated by dividing the total number of pulses 

output (nT) by the time between two rising edges of the flowmeter output (tTmeas) that occur 

immediately after the encoder rising edges that mark the start and stop of the piston travel time. 

Note that the encoder time is measured independently from the flowmeter time and while they 

are very nearly coincident, they are not necessarily equal in duration. As for the previous cases, 

the flowmeter pulse totalization time has quantization uncertainty of ∆t 6 . 

Figure 8. Diagram of the process for counting and timing the pulses from the encoder that 

measures piston displacement and from a pulse generating MUT (a turbine meter) and their 

quantization errors. 

The HLFS has redundancy in the encoders and oscillators in order to avoid miscounting pulses 

and to allow internal validation of measurements. The prover system uses two encoders (1 and 

2), each providing two pulse outputs (A and B). Output A originates from the leading edges of 

the encoder pulses and output B indicates the trailing edges. Therefore, a total of four 

chronometries (1A, 1B, 2A, and 2B) are used to measure the piston travel and thereby improve 

the accuracy of the measured collection time. Additionally, the HLFS uses two oscillators to 

measure the piston travel interval. The first oscillator is used to measure time for chronometries 

1A and 2A and the second oscillator operates on 1B and 2B. The Type B uncertainties of the 

clocks are assumed to be fully correlated between clocks operated by the same oscillator.

Increasing the number of chronometries or measurements does not improve the measurement 

uncertainty when the uncertainties are fully correlated. 

18
Fluid Properties

As indicated in (15), the thermal expansion of the fluid, and not the density itself, affects the 

volumetric flow determination. The physical property values involved in the expression for fluid 

density as a function of temperature are periodically determined off-line, using an oscillating 

tube densitometer (calibrated with distilled water and NIST Standard Reference Materials). A

sample set of measured fluid density and the thermal expansion coefficient are shown in Table 

B9, Appendix B. 

Likewise, the fluid kinematic viscosity does not directly affect the flow results in this type of

calibrator. However, depending on the type of MUT, the fluid kinematic viscosity can affect the 

flowmeter output. It has been shown that one can obtain improved flow measurement 

performance when calibration results are expressed in non-dimensional parameters, such as 

Strouhal and Roshko numbers [4]. Using these, variations in the fluid temperature, kinematic 

viscosity, and/or density, from those used during the calibration conditions, can be compensated 

for. By anticipating the use of these non-dimensional parameters, our uncertainty analysis should 

apply to a wide range of fluid kinematic viscosity (i.e., 0.5 centistokes to 2 centistokes). For the 

report of calibration, the fluid viscosities are periodically measured using a capillary viscometer 

(Schott AVS 440), which measures the time required for an amount of fluid to flow through a 

capillary tube of known diameter and length. Table B9, Appendix B shows a typical set of 

viscosity data. 

5.3 Propagating Components of Uncertainty 

Referring to the graphic representation of the uncertainty analysis, given in Fig. 5 and the 

uncertainty propagation equation (19) given above, the uncertainty of the sub-measurements 

must be assessed before the flow uncertainty can be estimated. This process is propagated 

throughout all the measurement components until the uncertainties of the desired quantities are 

obtained. The details of the calculations of the propagation calculations for the submeasurements are given in Appendix B. Table B2 is for the temperature measurement 

uncertainties, Table B6 is for diameter, piston displacement, and volume measurements, and 

Table B8 is for the encoder timing uncertainty. 

After the uncertainties of the sub-measurements are obtained, the one-standard-deviation, (k=1), 

uncertainty for the volumetric flow of 3.0 L/min (0.8 gal/min), as given in (15), is obtained and 

shown in Table 2. The second column in Table 2 lists the nominal mean values of the 

measurement components. The third and fourth columns list their Type A and Type B 

uncertainties, respectively. The fifth and sixth columns list the same uncertainties as a percent of

the mean. The seventh column lists the non-dimensional sensitivity coefficients of the 

component to the flow. The eighth and ninth columns list the flow uncertainty, in percent, due to 

the sub-component. These data show that the primary contribution to the flow uncertainty is the 

uncertainty of the piston displaced volume, ∆VP (0.0021 %) and to the temperature difference 

∆TP,MUT (-0.0031 %) while the uncertainty contribution due to the connecting volume uncertainty 

is very small (5.8E-5 %). The total uncertainties of the flow are 0.0007 % and 0.0037 % for the 

Type A and Type B, respectively. 

19
Table 2. Volumetric flow rate uncertainty propagation using (15). 

mean uA uB 

uA 

[%] 

uB 

[%] ciQ 

Qi uA

[%] 

Qi uB

[%] 

∆VP[cm3] 1514.9 0.0006 0.0317 3.9E-05 0.0021 1.00 3.9E-05 0.0021

t [s] 30.0 5.2E-06 3.3E-05 1.7E-05 1.1E-04 -1.00 -1.7E-05 -1.1E-04

α [1/K] 0.00098 0 2.7E-06 0 0.2768 -3.4E-05 0 -9.5E-06

αS [1/K] 1.7E-05 0 3.4E-07 0 2.00 -1.3E-07 0 -2.6E-07

VCV [cm3

] 195.5 0 22.73 0 11.6 4.9E-06 0 5.8E-05

∆TP,MUT [K] 0.040 0.001 0.031 3.24 77.5 -3.9E-05 -1.3E-04 -3.1E-03

∆TCV [K] 0.040 0.057 0 143 0 5.1E-06 7.3E-04 0

∆TCP [K] 0.020 0.049 0 243 0 -1.3E-07 -3.2E-05 0

QMUT [cm3

/s] 50.49 0.0004 0.0019 0.0007 0.0037

Using the same method, the uncertainties for other flows were calculated. Table 3 shows the 

results for a range of flows and the worst-case scenario for the total uncertainty for these flows. 

The worst-case scenario values are then used for the uncertainty specification for the 2 L HLFS: 

u A = 0.0018 % and uB = 0.0037 % (26)

Table 3. Uncertainties of the HLFS for several flows. 

Q c t ∆VP ∆L u A u B

[L/min] [s] [cm3] [cm] [%] [%]

0.2 180 608 15 0.0018 0.0037

0.4 180 1215 30 0.0009 0.0037

0.8 113 1515 37.4 0.0007 0.0037

1.5 60 1515 37.4 0.0007 0.0037

3.0 32 1515 37.4 0.0007 0.0037

5.0 18 1515 37.4 0.0007 0.0037

Max 0.0018 0.0037

5.4 Combined and Expanded Uncertainty 

20
In accordance with [8], the combined standard uncertainty for the measurement system, is given 

by 2 2

uC = u A + u B . That is: 

uC ( ) 0.0018 (0.0037) % 0.0041 % 2 2 = + = (27)

The approximate confidence level of the result given above is 68 %. When a coverage factor of 

 is used to convert the combined standard uncertainty to an expanded uncertainty, with an 

approximate 95 % level of confidence, the expanded uncertainty becomes: 

k = 2

U = ± k uC = 0.0082 % < 0.01% (28)

6.0 Uncertainty of Meter Under Test 

The uncertainty analysis reported above is for the respective piston-cylinder and associated 

connecting volume for the prover of the HLFS. The uncertainty analysis for the MUT results will 

depend on the flowmeter type and the associated instruments used. As indicated above, improved 

flow measurement performance can be obtained by using non-dimensional parameters. 

In addition to the system uncertainty discussed above, the uncertainty for the MUT should also 

include the standard deviation for replicated flow calibration results (i.e., reproducibility∗

) [1]. 

Reproducibility data is used because flowmeter users need to know the short term stability (i.e.,

repeatability◊

) and hysteresis of their instruments. The long term stability of the MUT is also 

important as are reproducibility when turned-off and turned-on, and the day to day changes in its 

performance. These types of reproducibility errors can be an order of magnitude larger than the 

MUT’s repeatability errors. The replicated uncertainty of the MUT is ascertained from multiple 

calibration results. By using this method, the total uncertainty of the calibration data for the test 

meter will include both contributions from the calibration system and MUT. 

 

∗ Reproducibility is defined as the closeness of the agreement between the results of measurements of the same measurand 

carried out under changed conditions of measurement [1].

◊ Repeatability is defined as the closeness of the agreement between the results of successive measurements of the same

measurand carried out under the same conditions of measurement [1].

 Hysteresis is defined as the closeness of the agreement between the results of measurements of the same measurand 

whether the value of the measurand is approached from a higher or lower values along its range (it is a form of reproducibility

[1])

21
7.0 References 

[1] ISO, “International Vocabulary of Basic and General Terms in Metrology,” 2nd ed., 

(International Organization for Standardization, Geneva, Switzerland, 1993). 

[2] G. E. Mattingly, “Flow Measurement Proficiency Testing,” ISA Flow Measurement, 2nd ed., 

Practical Guides for Measurement and Control, Ch. 28 (Instrument Society of America, NC, 

2001). 

[3] T. T. Yeh, P.I. Espina, G.E. Mattingly, and N.R. Briggs, “An Uncertainty Analysis of a 

NIST Hydrocarbon Liquid Flow Calibration Facility,” Proceedings of HT/FED’04, HTFED2004-56790, 2004 Heat Transfer/Fluids Engineering Summer Conference, Chariotte, 

North Carolina, USA, July 11-15, 2004. 

[4] G. E. Mattingly, “The Characterization of a Piston Displacement-Type Flowmeter Calibration Facility and the Calibration and Use of Pulsed Output Type Flowmeters,” NIST J. of

Res., 97 5 pp. 509-531, 1992. 

[5] J. D. Wright and G. E. Mattingly, “NIST Calibration Services for Gas Flow Meters: Piston 

Prover and Bell Prover Gas Flow Facilities”, NIST SP 250-49 (National Institute of Standards and Technology, Gaithersburg, MD, 1998). 

[6] API, “Manual of Petroleum Measurement Standards, Chapter 4.3: Proving Systems - Small 

Volume Provers”, 1st ed. (American Petroleum Institute, Washington, D.C., 1993). 

[7] B. N. Taylor and C. E. Kuyatt, “Guidelines for the Evaluating and Expressing the Uncertainty of NIST Measurement Results”, NIST TN-1297 (National Institute of Standards and 

Technology, Gaithersburg, MD, 1994). 

[8] ISO, “Guide to the Expression of Uncertainty in Measurement” (International Organization 

for Standardization, Geneva, Switzerland, 1993).

[9] Coleman, H. W. and Steele, W. G., “Experimentation and Uncertainty Analysis for 

Engineers,” 2nd ed., (John Wiley and Sons, 1999). 

22
Appendix A: Volume Balance Method 

An alternate method for correcting fluid volume in the connecting volume is the volume 

balance method. For a given fluid, the change of its volume, ∆VF , is given by the final fluid 

volume, V , minus the initial fluid volume, V , or, Ff Fi

∆VF = VFf −VFi (A1)

Using Figure A1 as a reference, the initial fluid volume can be expressed as, 

( ) Fi P CV MUT i V = V +V +V (A2)

while the final fluid volume is given by, 

( ) VFf = VP +VCV +VMUT f +VL (A3)

Figure A1. Balance of fluid quantities throughout the control volume. Different colors denote 

different fluid temperatures or different fluid densities. 

In (A2) and (A3), V and V are the fluid volumes contained in the piston prover at the 

initial (start) and final (stop) times, respectively. Similarly, V and V are the fluid volumes 

contained within the connecting pipe at the initial and final times, respectively. V and V

are the fluid volumes discharged through the MUT at the initial and final times, respectively, and 

 is the volume of any leaked fluid. As discussed above, the leakage V will be assumed to be 

zero and not considered further in this discussion. 

Pi Pf

CVi CVf

L

MUTi MUTf

VL

Combining (A1-A3), we obtain an expression for the net fluid volume discharged through 

the meter, 

∆VMUT = VMUTf −VMUTi = ∆VF + ∆VP − ∆VCV (A4)

23
where, is the net fluid volume displaced by the piston, and ∆VP = VPi −VPf ∆VCV = VCVf −VCVi

is the change of the connecting volume due to thermal expansion of the connecting pipe. Next, 

we shall discuss each of the volume change terms. 

The volume change of the connecting volume due to the temperature change, ∆ can be 

expressed as, 

TCP

∆VCV = VCV 3 α S ∆TCP (A5)

where V is the average connecting volume. CV

In contrast to , the volume change of the fluid, ∆VCV ∆VF , is more difficult to estimate. As 

seen before, because the modulus of elasticity of the fluid is very large ( 2 GPa) and the working 

pressure is small in comparison ( 80 kPa), and fairly constant (∆P < 80 Pa), the density of the 

liquid can be assumed to be function of fluid temperature only. Thus, for a given mass of fluid, 

changes in fluid volume will only be due to changes in fluid temperature. For a small change of

temperature∆T , it can be shown that: 

ρ = ρ 0 ( ) 1−α∆T , ∆ρ = −ρ 0α∆T , and V = V0α∆T (A6)

If the fluid volume is divided into small volume elements, V (see Figure 5), such that each 

small volume has constant fluid temperature, T , the volume change of each element can be 

expressed as, 

j

j

( ) j j j j P ∆V =V α∆T =α ∆H / ρ c (A7)

where is the heat transferred to the -th element of the fluid, and c is the specific heat 

capacity of the fluid. By summing over all elements, the total (or net) volume increase of the 

fluid is, 

∆H j j P

∆VF = ∑∆V j = ( ) α / ρ cP ∑∆H j =(α / ρ cP )∆H (A8)

where ∆H is the total (or net) heat addition to the fluid in the connecting volume during the 

calibration time . c t

From (A8), it is worth noting that heat transfer between cold and hot pockets of fluid does 

not change the total volume of the fluid. Only net heat addition to the fluid will change the total 

fluid volume. Thus, if there is no net heat addition to the fluid, there will be no change in the 

fluid volume. 

In our calibrator, some heat addition is inevitable due to viscous dissipation as the fluid 

moves through the pipe. There is also some heat transfer between the pipe and fluid. Thus, the 

net heat addition is normally a function of flow and the temperature difference between the 

calibrator and the ambient air. Insulating the connecting pipe can reduce the heat transfer to or 

from between the room, thus normally reducing the amount of heat transfer between the pipe and 

fluid. Also, operating the calibrator near room temperature will reduce this heat transfer. 

Equation (A8) further shows that, large temperature coefficients, α , will lead to large volume 

increases, while large fluid densities, ρ , or specific heat capacities, c , will result in smaller 

volume increases. 

P

24
According to the conservation of energy, the heat addition, ∆H for the liquid is related to the 

average temperature rise of the fluid as it moves through the connecting volume, , as: ∆TH ,CV

P VP TH ,CV ∆H = ρ c ∆ ∆ (A9)

Under normal operational conditions, the heat addition is a system characteristic which should be 

fairly constant. 

By combining (A8) and (A9) and eliminating the heat addition, ∆H , the fluid volume 

change, ∆VF , becomes: 

∆VF = ∆VPα ∆TH ,CV (A10)

This equation shows that the fluid volume increase depends only on the temperature rise, 

, and the thermal expansion coefficient of the fluid, ∆TH ,CV α , but not on the fluid specific heat 

capacity explicitly. 

By combining (A4), (A5) and (A10) and dividing by the calibration time, , we obtain the 

average fluid volume flowing through the MUT. That is, 

ct

[ ] P H ,CV CV S CP

c c

MUT

MUT V ( T ) V T

t t

V Q = ∆ + ∆ − ∆ ∆ = 1 α 3 α

1

(A11)

Once again, it is worth noting that the fluid density does not directly affect the volumetric flow 

calculation. Next, we will discuss how we determine the fluid temperature rise in the connecting 

volume during the calibration. 

The fluid passing through the MUT is not exactly the same fluid that is being discharged 

from the prover. For the sake of discussion, the fluid that affects the average temperature rise can 

be divided into three parts: I, II and III. Figure A2 shows the sketch of the distribution of the 

three parts of fluid and the temperature variations as functions of time at MUT, TMUT, and at 

piston, . Part I fluid is the fluid found in the connecting volume at the time the calibration 

starts. During the calibration period, this fluid is pushed through the MUT. All of the fluid in 

Part II is displaced by the piston and passes through the MUT during the calibration period. The 

fluid in Part III is also displaced by the piston but does not reached the MUT before the calibration ends (i.e., still in the connecting volume). The determination of the temperature rise for the 

fluid in part II is simple, while those for part I and III are more complex. The average temperature rise, ∆ is thus the weighted average temperature rise of the three parts of fluid. 

TP

TH ,CV

For each fluid element, its temperature rise is the difference of its temperature at the initial 

and final times. To help illustrate this point, the fluid temperature at position x and time is 

given as T . Here, 

t

( x,t ) x is the distance measured from the piston, i.e., at the piston and 

 at the MUT. The temperature at two special locations, the MUT and the piston, 

(shown on Figure A2) are: 

x = 0

MUT x = x

TMUT (t ) = T( xMUT ,t ) and T (t ) T( ,t ) P = 0 (A12)

and the temperatures at two special times, the initial and final times, are: 

T ( x ) T( x, ) i = 0 and T ( x ) T( x,t ) f = c (A13)

The position of a fluid element at time t, x, can be expressed by 

25
x x U t = 0 + CV (A14)

where is the initial position of the fluid, and U is the average fluid 

velocity in the connecting volume. Here, is the inner diameter of the connecting pipe. 

0 x

2 4 CV MUT CV = Q / π d

dCV

Figure A2. Sketch and estimation of the average temperature rise of fluid as it moves through 

the connecting volume. UCV is the average flow velocity in the connecting volume. TMUT(t) and 

TP(t) are the temperature as functions of time at the MUT and at the prover, respectively. 

As referred to Figure A2, we have the average temperature rise 





 



 ∆ = ∆ + ∆ + ∆ ∫ ∫ ∫ −

c

c r

c

r

r t

t t III

t

t II

t

I

c

H ,CV T dt T dt T dt

t

T

0

1

(A15)

where 

T T (t ) T( x U t, ) I MUT MUT CV ∆ = − − 0 (A16)

is the temperature rise of the fluid element I located at the MUT and time t ; 

26
T T (t ) T (t t ) ∆ II = MUT − P − r (A17)

is the temperature rise of the fluid element II; and 

T T(U (t t ),t ) T (t ) ∆ III = CV c − c − P (A18)

is the temperature rise of the fluid element III located at the Piston and time t . Inserting 

(A16~A18) into (A15), we have 

[

[ ]

( )

CV

P

CP

P,MUT

CV,f CV,i

c

r P,MUT

CV

x

CV

x

c

c

MUT,m P,m

t

t t CV c c

t

MUT CV

t

P

t

MUT

c

t

t t CV c c P

t

t MUT CV MUT P r

t

MUT

c

H,CV

T

V

V T

T T

t

t T

U

dx T( x, ) Udx T( x,t ) t

T T

T( x U t, )dt T(U (t t ),t )dt

T (t )dt T (t )dt

t

T(U (t t ),t ) T (t )dt

[T (t ) T( x U t, )]dt T (t ) T (t t

t

T

MUT MUT

c

c r

r

c c

c

c r

c

r

r

∆

∆ = −∆ +

= −∆ + −





 



 = − + −

























− − + −

−

=













+ − −

− − + − −

∆ =

∫ ∫

∫ ∫

∫ ∫

∫

∫ ∫

−

−

0 0

0

0 0

0

0 1

0

1

0 1

(A19)

Thus, when substituting (A19) into (A11), we obtain the average fluid volume flowing through 

the MUT as, 

[ ] V ( T ) V ( T T ) t

Q P P,MUT CV CV S CP

c

MUT = ∆ 1−α∆ + α ∆ −3α ∆ 1

(A20)

This equation is exactly (15), which was obtained from the mass conservation method. 

27
Appendix B: Supporting Data for Uncertainty Analysis 

B1. Temperature measurement uncertainty components. 

Table B1. Thermistor calibration uncertainties: the standard deviation of repeated readings (uA) 

and the difference from the reference sensor (uB). The maximum values of all sensors (TUU4A) 

were used for the uncertainties of the thermistors; uA = 0.041 K and uB = 0.031 K. 

T-Tref

uA uB

[K] [K]

TP1A 0.007 -0.015

TP2A 0.026 0.010

TUU4A 0.041 0.031

TCV1B 0.031 0.006

TCV1A 0.017 -0.005

TUU4B 0.032 -0.012

TP1B 0.010 -0.012

TP2B 0.009 -0.011

TCV2 0.011 -0.014

TCV3 0.004 0.000

TAir 0.029 -0.019

TAthena 0.027 -0.019

Max 0.041 0.031

 Table B2. Temperature measurement uncertainties. 

TPx,m n>=1000, x=A or B 

mean n uA uB 

TPx [K] 295.2 1000 0.041 0.031

TPx,m [K] 295.2 0.001 0.031

TMUTx,m n>=1000, x=A or B 

mean n uA uB 

TMUTx [K] 295.2 1000 0.041 0.031 

TMUTx,m [K] 295.2 0.001 0.031

TAir,m n>=1000 

mean n uA uB 

TAir [K] 295.2 1000 0.041 0.031 

28
TAir,m [K] 295.2 0.001 0.031 

TSA Temperature sensors used for spatial average 

mean ci uA uB 

Ti [K] 295.2 1 0.041 0.031

EMax [K] 0 1 0.1 0 

TSA [K] 295.2 0.108 0.031 

TP,x=(TPA+TPB)/2 , x=i or f

mean ci uA uB 

TPA,x [K] 295.2 0.5 0.108 0.031 

TPB,x [K] 295.2 0.5 0.108 0.031 

TP,x [K] 295.2 0.076 0.022 

TP,m=(TPA,m+TPB,m)/2 

mean ci uA uB 

TPA,m [K] 295.2 0.5 0.001 0.031 

TPB,m [K] 295.2 0.5 0.001 0.031 

TP,m [K] 295.2 0.001 0.022 

TMUT,x=(TMUTA+TMUTB)/2 , x=i or f

mean ci uA uB 

TMUTA,x [K] 295.2 0.5 0.108 0.031 

TMUTB,x [K] 295.2 0.5 0.108 0.031 

TMUT,x [K] 295.2 0.076 0.022 

TMUT,m=(TMUTA,m+TMUTB,m)/2 

mean ci uA uB 

TMUTA,m [K] 295.2 0.5 0.001 0.031 

TMUTB,m [K] 295.2 0.5 0.001 0.031 

TMUT,m [K] 295.2 0.001 0.022 

TCV1=(TCV1A+TCV1B)/2 

mean ci uA uB 

TCV1A [K] 295.2 0.5 0.108 0.031 

TCV1B [K] 295.2 0.5 0.108 0.031 

29
TCV1 [K] 295.2 0.076 0.022 

∆TP,MUT = TP,m -TMUT,m

mean ci i uA uB 

TP,m [K] 295.2 1 0.001 0.022 

TMUT,m [K] 295.2 -1 0.001 0.022

∆TP,MUT [K] 0.040 0.001 0.031 

TCV,x=(TP,x+TCV1+TCV2+TCV3+TMUT,x)/5 , x=i or f

mean ci uA uB 

TP,x [K] 295.2 0.20 0.076 0.022

TCV1 [K] 295.2 0.20 0.076 0.022

TCV2 [K] 295.2 0.20 0.108 0.031

TCV3 [K] 295.2 0.20 0.108 0.031

TMUT,x[K] 295.2 0.20 0.076 0.022

TCV,x [K] 295.2 0.040 0.012 

∆TCV = TCV,f-TCV,i

mean ci uA uB 

TCV,f [K] 295.2 1 0.040 0.012

TCV,i [K] 295.2 -1 0.040 0.012 

∆TCV [K] 0.040 0.057 0.000 

TCP,x=(TP,x+TCV1+TCV2+TCV3+TMUT,x+TAir)/6, x=i or f

mean ci uA uB 

TP,x [K] 295.2 0.1667 0.076 0.022

TCV1 [K] 295.2 0.1667 0.076 0.022

TCV2 [K] 295.2 0.1667 0.108 0.031

TCV3 [K] 295.2 0.1667 0.108 0.031

TMUT,x [K] 295.2 0.1667 0.076 0.022

TAir [K] 295.2 0.1667 0.041 0.031

TCP,x [K] 295.2 0.034 0.011 

30
∆TCP = TCP,f-TCP,i

mean ci uA uB 

TCP,f [K] 295.2 1 0.034 0.011 

TCP,i [K] 295.2 -1 0.034 0.011 

∆TCP [K] 0.020 0.049 0.000 

∆TDR = TP,m -TRD

mean ci uA uB 

TP,m [K] 295.2 1 0.001 0.022 

TRD [K] 295.3 -1 0.010 0.010 

∆TDR [K] -0.08 0.010 0.024 

∆TdR = TP,m -TRd

mean ci uA uB 

TP,m [K] 295.2 1 0.001 0.022 

TRd [K] 295.4 -1 0.010 0.010 

∆TdR [K] -0.21 0.010 0.024 

∆TER = TAir,m -TRE

mean ci uA uB 

TAir,m [K] 295.2 1 0.001 0.031 

TRE [K] 295.5 -1 0.010 0.010 

∆TER [K] 1.000 0.010 0.033 

31
B2. Diameter, piston displacement, and volume measurement uncertainties. 

Table B3. Shaft Diameter, dR and its uncertainty. 

Mitutoyo Outside diameter micrometer Model: MDC-2"JT, s/n:293-370 

Standard reference Blocks calibrated by NIST Precision Engineering Division 

 Test 1 Test 2 Test 3 Test 4 Average 

 T [C] 22.353 22.310 22.396 22.564 22.406

position # position 

[-] [cm] [mm] [mm] [mm] [mm]

1 56.5 25.4446 25.4425 25.4437 25.4424

2 51.6 25.4428 25.4431 25.4432 25.4440

3 46.5 25.4431 25.4435 25.4439 25.4437

4 41.5 25.4402 25.4405 25.4406 25.4416

5 36.5 25.4413 25.4419 25.4418 25.4450

6 31.5 25.4400 25.4406 25.4413 25.4412

7 26.5 25.4412 25.4411 25.4403 25.4413

8 21.5 25.4399 25.4406 25.4414 25.4416

9 16.5 25.4402 25.4404 25.4405 25.4413

10 11.5 25.4399 25.4403 25.4404 25.4414

11 6.5 25.4384 25.4343 25.4391 25.4388

 mean dR 25.4411 25.4408 25.4415 25.4420 25.4413 

 std unc 5.38E-04 7.34E-04 4.76E-04 5.04E-04 0.0006 

 Ref gage unc 6.25E-05 

 Combined unc 0.00057 

32
Table B4. Cylinder inside diameter, DR and its uncertainty. 

Mitutoyo 3-legged inner diameter micrometer, Model: IT-005D, s/n=10019027 

Two Standard reference Rings calibrated by NIST Precision Engineering Division 

 Test 1 Test 2 Test 3 Average 

Position T [C] 22.199 22.189 22.464 22.284 

# position Gap Gap Gap

[-] [cm] [mm] [mm] [mm]

1 59.182 -0.01711 -0.01931 -0.01632

2 54.182 -0.01751 -0.01992 -0.01642

3 49.182 -0.01912 -0.01992 -0.01722

4 44.182 -0.01982 -0.02092 -0.01832

5 37.182 -0.01501 -0.01631 -0.01502

6 30.182 -0.01481 -0.01681 -0.01562

7 35.858 -0.01462 -0.01683 -0.01842

8 28.858 -0.01322 -0.01643 -0.01752

9 21.858 -0.01882 -0.02064 -0.02192

10 16.858 -0.01872 -0.02024 -0.02102

11 11.858 -0.01702 -0.01884 -0.01932

12 6.858 -0.01812 -0.02024 -0.02162

Mean gap -0.0170 -0.0189 -0.0182 -0.0180 

std unc 0.00061 0.00051 0.00067 0.00060 

Ref ring 76.2056 76.2056 76.2056 

Mean DR 76.1886 76.1867 76.1873 76.1875 

 Ref unc 0.00015 

 Combined unc 0.00062 

33
Table B5. Encoder constant, KER and its uncertainty. 

Standard Laser Interferometer, HP Model#10565A, NIST Precision Engineering Division 

Ref unc = 6.25E-5 mm

Ref, dL 1a 1b 2a 2b K, 1a K,1b K,2a K,2b average

mm Pulse Pulse Pulse Pulse p/mm p/mm p/mm p/mm p/mm

551.813 27586 27586 27598 27596 49.9916 49.9916 50.0134 50.0097

550.370 27514 27514 27524 27524 49.9918 49.9918 50.0100 50.0100

549.345 27462 27462 27475 27473 49.9905 49.9905 50.0141 50.0105

550.951 27543 27543 27553 27553 49.9918 49.9918 50.0099 50.0099

mean KER 49.9914 49.9914 50.0118 50.0100 50.001

 std unc 0.0003 0.0003 0.0011 0.0002 0.0005 

Ref unc 6.3E-06

 Combined unc 0.0005 

Table B6. Diameter, piston displacement, and volume measurement uncertainties. 

D =DR(1+αs ∆TDR) 

mean uA uB %uA %uB ciD D %uA D %uB 

DR [cm] 7.6188 0.0E+00 0.0001 0.0E+00 0.0008 1.0000 0.0E+00 0.0008 

αs [1/K] 1.7E-05 0.0E+00 3.4E-07 0.0E+00 2.0000 -1.4E-06 0.0E+00 -2.9E-06

∆TDR [K] -0.084 0.010 0.024 -11.955

-

28.6828 

-1.4E06 1.7E-05 4.1E-05

D [cm] 7.6188 1.3E-06 0.0001 1.7E-05 0.0008 

d =dR(1+αs ∆TdR) 

mean uA uB %uA %uB cid d %uA d %uB 

dR [cm] 2.5441 0.0E+00 0.0001 0.0E+00 0.0022 1.00000 0.0E+00 0.0022 

αs [1/K] 1.7E-05 0.0E+00 3.4E-07 0.0E+00 2.0000 

-3.5E06 0.0E+00 -7.0E-06

∆TdR [K] -0.206 0.010 0.024 -4.875

-

11.6959 

-3.5E06 1.7E-05 4.1E-05

d [cm] 2.5441 4.3E-07 0.0001 1.7E-05 0.0022 

34
KE=KR/(1+ αE ∆TER) 

mean uA uB %uA %uB ciK K %uA K %uB 

KR[p/cm] 500.01 0.0E+00 0.0048 0.0E+00 0.0010 1.0000 0.0E+00 0.0010

αE [1/K] 8.0E-06 0.0E+00 1.0E-06 0.0E+00 12.5000 

-8.0E06 0.0E+00 -1.0E-04

∆TER [K] 1.000 0.010 0.033 1.008 3.257

-8.0E06 -8.1E-06 -2.6E-05

KE [p/cm] 500.02 4.0E-05 0.0048 8.1E-06 0.0010 

∆Vp= π (D2

-d2)/4L = π (D2-d2)/4NE/KE

mean uA uB %uA %uB ciV V %uA V %uB 

D [cm] 7.6188 1.3E-06 0.0001 0.0000 0.0008 2.25101 3.8E-05 0.0018

d [cm] 2.5441 4.3E-07 0.0001 0.0000 0.0022 0.25101 -4.3E-06 -0.0006

KE [p/cm] 500.02 4.0E-05 0.0048 8.1E-06 0.0010 -1.000 -8.1E-06 -0.0010

∆VP [cm3

] 1514.88 0.0006 0.0317 3.9E-05 0.0021

VCV=πdCV

2

/4lCV

mean uA uB %uA %uB ciV V %uA V %uB 

dCV [cm] 0.9525 0.0E+00 0.0508 0.0E+00 5.3333 2 0.0E+00 10.6667

lCV [cm] 274.32 0.0E+00 12.700 0.0E+00 4.6296 1 0.0E+00 4.6296

Vcv [cm3

] 195.47 0.0E+00 22.729 0.0E+00 11.6280

35
B3. Timing and counter uncertainties.

Table B7. Sample calibration data for two oscillators that are used for time and frequency 

measurements. 

HP Universal counter 53131A 

S/N: Ref #1 =3736A22409, Ref #2 =3736A22423 

Standard frequency: 1 MHz, uA=0.2 Hz, uB=1 Hz 

Oscillator #1 Oscillator #2 

counter #1 counter #2 average counter #1 counter #2 average 

[Hz] [Hz] [Hz] [Hz] [Hz] [Hz]

999998.798 999998.777 999998.788 1000005.887 1000005.812 1000005.85

999998.792 999998.778 999998.785 1000007.224 1000005.231 1000006.23

999998.793 999998.770 999998.782 1000006.115 1000005.038 1000005.58

999998.736 999998.763 999998.750 1000005.915 1000005.940 1000005.93

999998.791 999998.781 999998.786 1000005.193 1000005.916 1000005.55

999998.794 999998.778 999998.786 1000005.193 1000005.914 1000005.55

999998.781 999998.780 999998.781 1000005.193 1000005.911 1000005.55

999998.785 999998.782 999998.784 1000005.193 1000005.880 1000005.54

999998.782 999998.781 999998.782 1000005.193 1000005.928 1000005.56

999998.791 999998.789 999998.790 1000005.193 1000005.940 1000005.57

 avg 999998.781 1000005.69 

uA st dev 0.011 0.235 

uB uncertainties due to temperature effects 

 T avg T avg 

 [C] [Hz] [C] [Hz] 

 24.0 999998.781 24.0 1000006.001 

 24.0 999998.786 24.0 1000005.690 

 26.0 999998.761 26.0 1000005.966 

 22.5 999998.761 22.5 1000006.934 

 22.5 999998.757 22.5 1000006.841 

 19.5 999998.761 19.5 1000007.570 

Temp consatant, Hz/C 0.00177 -0.2972 

Max. diff. on T, C 4 4 

 uB 0.0071 -1.1887 

Combined uncertainties 

 uA , [× 10-6] uB, [× 10-6] uA , [× 10-6] uB, [× 10-6]

 0.20 1.00 0.31 1.55 

worst case uncertainty selected for the oscillator uncertainty 

 uA , [× 10-6] uB, [× 10-6] 

 0.31 1.55 

36
Table B8. Encoder timing uncertainty. 

t=(t1A+t1B+t2A+t2B)/4 

mean uA uB %uA %uB cit t %uA t %uB 

t1A [s] 30.000 1.0E-05 4.7E-05 3.4E-05 1.6E-04 0.25 8.6E-06 3.9E-05

t1B [s] 30.000 1.0E-05 4.7E-05 3.4E-05 1.6E-04 0.25 8.6E-06 3.9E-05

t2A [s] 30.000 1.0E-05 4.7E-05 3.4E-05 1.6E-04 0.25 8.6E-06 3.9E-05

t2B [s] 30.000 1.0E-05 4.7E-05 3.4E-05 1.6E-04 0.25 8.6E-06 3.9E-05

t [s] 30.000 5.2E-06 3.3E-05 1.7E-05 1.1E-04

Table B9. Properties of hydrocarbon liquid fluid, MIL-C-7024C. 

Fluid density Fluid kinematic viscosity 

T ρ T ν 

[C] [kg/m3] [C] [cs] 

15.14 788.0 20.03 1.248

19.98 784.4 20.03 1.254

24.95 780.5 24.99 1.164

29.70 776.8 24.99 1.164

 29.95 1.088 

 29.95 1.089 

 29.95 1.088 

 29.95 1.066 

 29.95 1.061 

 15.08 1.344 

 15.08 1.344 

 

ρ =ρo(1+α ∆T) ν =νo + c T 

ρo α Tr νo c 

[kg/m3] [1/C] [C] [cs] [cs/C] 

784.3 0.0010 20 1.610 -0.018

37
Appendix C: Sample Calibration Report 

REPORT OF CALIBRATION 

For A Dual Turbine Flowmeter 

April 28, 2005 

 

Manufacturer: Turbine Meters Inc. 

Serial Number: ABC123 

submitted by 

Customer Co. Name

Gaithersburg, MD 

(Purchase Order Number 123456, dated April 1, 2005) 

The flowmeter identified above was calibrated using the NIST 2 L Hydrocarbon Liquid Flow 

System (HLFS). This facility calculates volumetric flow by measuring the time required for a 

piston to displace a certain volume within a cylinder. Averages of the output of the meter under 

test were gathered over the same intervals that the volumetric flow was calculated from the 

primary standard. 

A dimensionless presentation of the calibration data facilitates accurate flow measurements by 

the flowmeter user even when the conditions of usage (liquid type, temperature) differ from the 

conditions during calibration. For a turbine flowmeter, the suggested parameters are the 

NIST Test Number: 836/999999-05 Page 38 of 44 

Calibration Date: July 15, 2005 by Technician’s Name and Project Leader’s Name
REPORT OF CALIBRATION Dual Turbne Flowmeter

Customer Company Name. Purchase Order No. A1234

dimensionless meter factor1

, i.e., the Strouhal number, St, and for the meter frequency, the 

Roshko number, Ro. These are written, 

 

M

3 3

M 4Q

f d

4

K d

U

f d St π π = = = , and (1) 

 ν

2 f d Ro = , (2)

where, in compatible units: 

 QM

f K = , the meter factor in pulses per unit volume, 

 d = the diameter of the meter (nominal diameter as specified by the manufactuere) 

 QM = volumetric flow, 

 2

M

M d

4Q

π

U = ,average flow velocity in the meter,

 f = the frequency of the meter output signal, and 

 ν = the kinematic viscosity of the fluid. 

The test liquid is MIL-C-7024C fluid (also know as Stoddard or Type 2 solvent) – a surrogate 

liquid for JP-4 and JP-5 jet fuels. The densities and viscosities of the liquid used in the system

were calibrated by NIST to have the following dependencies on temperature, assuming pressure 

effects to be negligible: 

ρ = 782.614 *(1 - 0.000984 * (T - 22.2)), and (3) 

ν

 

= 1.6097 - 0.01777 * T , (4) 

where, 

 ρ = the fluid density, in kg/m3, 

 ν = the kinematic viscosity of the fluid, in centistokes, and 

1

 G. E. Mattingly, “The Characterization of a Piston Displacement-Type Flowmeter Calibration Facility and the Calibration and Use of

Pulsed Output Type Flowmeters,” NIST J. of Res., 97 5 (1992)

NIST Test Number: 836/999999-05 Page 39 of 44 

Calibration Date: July 15, 2005 by Technician’s Name and Project Leader’s Name
REPORT OF CALIBRATION Dual Turbne Flowmeter

Customer Company Name. Purchase Order No. A1234

 T = the temperature of the fluid, in o

C. 

The diameter of the flowmeter used in calculations of the Ro and St is 12.5 mm, the nominal 

diameter specified by the manufacturer. The dual turbine flowmeter was installed in a test section 

with 100 diameters of straight pipe upstream and 75 diameters of straight pipe downstream. A

photograph of the installation is shown in Figure 1. 

Figure 1 A photograph of the flowmeter installation. 

An analysis was performed to assess the uncertainty of the calibration results obtained for the 

meter under test.2, 3, 4 The process involves identifying the equations used in calculating the 

calibration result (measurand) so that the sensitivity of the result to uncertainties in the input 

quantities can be evaluated. The approximately confidence level uncertainty of each of the 

input quantities is determined, weighted by its sensitivity, and combined with the other 

uncertainty components by root-sum-square to arrive at a combined uncertainty (U ). The 

combined uncertainty is multiplied by a coverage factor of 2.0 to arrive at an expanded 

uncertainty (U ) of the measurand with approximately 95 confidence level. 

67%

c

e %

As described in the references, if one considers a generic basis equation for the measurement 

process, which has an output, y , based on N input quantities, , i x

 

2 International Organization for Standardization, Guide to the Expression of Uncertainty in 

Measurement, Switzerland, 1996 edition.

3 Taylor, B. N. and Kuyatt, C. E., Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement

Results, NIST TN 1297, 1994 edition.

4 Coleman, H. W. and Steele, W. G., Experimentation and Uncertainty Analysis for Engineers, John Wiley and 

Sons, 2nd ed., 1999. 

NIST Test Number: 836/999999-05 Page 40 of 44 

Calibration Date: July 15, 2005 by Technician’s Name and Project Leader’s Name
REPORT OF CALIBRATION Dual Turbne Flowmeter

Customer Company Name. Purchase Order No. A1234

( , , , ) 1 2 N y = y x x K x (1)

and all uncertainty components are uncorrelated, the normalized expanded uncertainty is given 

by, 

( ) ( ) ( ) ∑= 













 = =

N

i i

i

i

e c

x

u x k s

y

U y k

y

U y

1

2

2 (2)

In the normalized expanded uncertainty equation, the u are the standard uncertainties of 

each input, and are their associated sensitivity coefficients, given by, 

( x )' s i

s ' s i

y

x

x

y s i

i

i ∂

∂ = (3)

The normalized expanded uncertainty equation is convenient since it permits the usage of 

relative uncertainties (in fractional or percentage forms) and of dimensionless sensitivity 

coefficients. The dimensionless sensitivity coefficients can often be obtained by inspection since 

for a linear function they have a magnitude of unity. 

For this calibration, the uncertainty of the Strouhal number has components due to the 

measurement of the actual volumetric flow at the meter under test u(QM ), the frequency output 

of the meter under test U( f ), and the reproducibility of the test u(R), all of which have 

sensitivity coefficients of 1. The expanded uncertainty of the volumetric flow from the HLFS has 

uncertainty of 0.001 %5

 which leads to a standard uncertainty of 0.00 5%. 

The relative standard uncertainty of the frequency measurements is less than 2 ×10-6 based on 

analysis of calibration records and can be considered negligible. The present uncertainty analysis 

does not include an uncertainty from the nominal diameter of the MUT.

 

5 T. T. Yeh, P.I. Espina, G.E. Mattingly, and N.R. Briggs, An UncertaintyAnalysis of a NIST Hydrocarbon Liquid 

Flow Calibration Facility, Proceedings of HT/FED’04, HT-FED2004-56790, 2004 Heat Transfer/Fluids 

Engineering Summer Conference, Charlotte, North Carolina, USA, July 11-15, 2004. 

NIST Test Number: 836/999999-05 Page 41 of 44 

Calibration Date: July 15, 2005 by Technician’s Name and Project Leader’s Name
REPORT OF CALIBRATION Dual Turbne Flowmeter

Customer Company Name. Purchase Order No. A1234

A normal flow calibration performed by the NIST Fluid Metrology Group consists of six flows 

spread over the range of the flowmeter. A flowmeter is normally calibrated at 10 %, 20 %, 35%, 

50 %, 75 %, and 100 % of the meter full scale. At each of these flow set points, five flow 

measurements are made. The same set point flows are tested on a second occasion, but the flows 

are tested in decreasing order instead of the increasing order of the first set. Therefore, the final 

data set consists of ten primary flow measurements made at six flow set points, i.e., 60 individual 

flow measurements. To measure the reproducibility6

 of the test, the standard deviation of the 

Strouhal number at each of the nominal flows was used to calculate the relative standard 

uncertainty (the standard deviation divided by the mean and expressed as a percentage). 

Using the values given above results in the expanded uncertainties listed in the following data 

tables and shown as error bars in the figures. Data for the upstream rotor (turbine 1) and the 

downstream rotor (turbine 2), as well as for their sum (turbine 1&2) are shown. 

Turbine1

1.690

1.695

1.700

1.705

1.710

1.715

0 5000 10000 15000 20000 25000 30000

Ro

St

d1 d2 avg

Figure 2. Strouhal number versus Roshko number for the upstream rotor (turbine 1). 

Symbols “d1” and “d2” represent the averages of data collected on two different occasions, 

while “avg” is the results for Day-1&2, reproducibility. 

 

6 Reproducibility is herein defined as the closeness of agreement between measurements with the flow changed and 

then returned to the same nominal value.

NIST Test Number: 836/999999-05 Page 42 of 44 

Calibration Date: July 15, 2005 by Technician’s Name and Project Leader’s Name
REPORT OF CALIBRATION Dual Turbne Flowmeter

Customer Company Name. Purchase Order No. A1234

Turbine2

1.770

1.775

1.780

1.785

1.790

1.795

1.800

1.805

0 5000 10000 15000 20000 25000 30000

Ro

St

d1 d2 avg

Figure 3. Strouhal number versus Roshko number for the downstream rotor (turbine 2). 

Symbols “d1” and “d2” represent the averages of data collected on two different occasions, 

while “avg” is the results for Day-1&2, reproducibility. 

combined

1.720

1.730

1.740

1.750

1.760

1.770

1.780

0 5000 10000 15000 20000 25000 30000

Ro

St

d1 d2 avg

Figure 4. Strouhal number versus Roshko number for the sum of the upstream and downstream

rotor (turbine 1&2). Symbols “d1” and “d2” represent the averages of data collected on two 

different occasions, while “avg” is the results for Day1&2, reproducibility. 

Table 1. Statistical Results and Uncertainties for dual turbine flowmeter.

NIST Test Number: 836/999999-05 Page 43 of 44 

Calibration Date: July 15, 2005 by Technician’s Name and Project Leader’s Name
REPORT OF CALIBRATION Dual Turbne Flowmeter 

Customer Company Name. Purchase Order No. A1234

NIST Test Number: 836/999999-05 Page 44 of 44 

Calibration Date: July 15, 2005 by Technician’s Name and Project Leader’s Name

TM

[

o

C] 

 

[kg/L] 



[cm2

/s] 

QM

[L/min] 

f 

[Hz] 

Ro

[d2

f/ 

St

[d2

f/4QM] 

Std Dev 

[%] 

Uk=2 

[%] 

Turbine 1 

22.42 782.8 0.0121 1.518 212.96 7089.2 1.6932 0.0244 0.0487 

22.39 782.8 0.0121 2.271 321.25 10689.6 1.7071 0.0740 0.1480 

22.35 782.7 0.0121 3.051 432.16 14370.0 1.7092 0.0955 0.1910 

22.40 782.8 0.0121 3.814 539.99 17969.1 1.7085 0.0754 0.1508 

22.40 782.8 0.0121 4.573 647.23 21537.9 1.7078 0.0159 0.0317 

22.44 782.8 0.0121 5.311 751.93 25037.0 1.7084 0.0144 0.0288 

Turbine 2 

22.42 782.8 0.0121 1.518 226.36 7535.1 1.7997 0.1567 0.3134 

22.39 782.8 0.0121 2.271 335.19 11153.4 1.7811 0.1024 0.2048 

22.35 782.7 0.0121 3.051 449.14 14934.6 1.7764 0.0903 0.1805 

22.40 782.8 0.0121 3.814 561.67 18690.5 1.7771 0.1135 0.2271 

22.40 782.8 0.0121 4.573 674.26 22437.4 1.7791 0.0856 0.1712 

22.44 782.8 0.0121 5.311 783.81 26098.5 1.7808 0.0639 0.1278 

Turbine 1&2 

22.42 782.8 0.0121 1.518 219.66 7312.2 1.7464 0.0884 0.1767 

22.39 782.8 0.0121 2.271 328.22 10921.5 1.7441 0.0874 0.1749 

22.35 782.7 0.0121 3.051 440.65 14652.3 1.7428 0.0927 0.1855 

22.40 782.8 0.0121 3.814 550.83 18329.8 1.7428 0.0929 0.1857 

22.40 782.8 0.0121 4.573 660.74 21987.7 1.7435 0.0502 0.1005 

22.44 782.8 0.0121 5.311 767.87 25567.7 1.7446 0.0377 0.0755 

For the Director, 

National Institute of Standards and Technology 

Group Leader, Fluid Metrology Group 

Process Measurements Division 

Chemical Science and Technology Laboratory